From the point of view of proof-theoretic semantics the following two points are essential: (1) Introduction rules (clauses) for logically compound formulas are not distinguished in principle from introduction rules (clauses) for atoms.
In view of the proof of Theorem 3.1, we only need to emphasis those steps where the proofs differ.
In view of the proof of Theorem 2.1, it holds that ρ 2 ≥ ρ 1. Similarly as in the proof of Theorem 2.1, it follows from (2.4), (2.7), (2.8), (2.9), and (2.10) that there exists a real number C > 0 such that | c m | ≤ C | a m | (3.3).
Moreover, if x = S x, then p ( x, x ) = 0. Proof Observe that the sequence { S n x 0 } is a Cauchy sequence (in view of the proof of Theorem 1), and so there exists a point x ∗ in X such that lim n → ∞ S n x 0 = x ∗.
Proof Let ε q > 0, q ∈ N, be a sequence converging to 0. In view of the proof of the theorem, for each q ∈ N there exists a point p q ∈ S 2 such that z i ( p q ) ≤ ε q, i ∈ [ n ].
In view of the proof of Theorem 3.1, we can get the population mean.
In view of the proof of Theorem 4.1, the operator Υ maps E into itself.
In view of the proof of Theorem 3.1, we show the difference only.
In view of the proof of Theorem 3.1, we show that u = ū.
In view of the proof of Case (C3), we have (2.16) holds.
end{aligned} In view of the proof above, we get the following.
